How do I fix this strange effect that happens when I start animating items that are set to cloth in physics?
The problem is that your Solidify modifier is above the Cloth modifier:



This means the cloth simulation will use that double layer of geometry, which can cause some instability and self-intersections.

In your case, the inside layer is poking through the outside layer and causing shading issues due to the fact that the normals of the new outside layer are facing the wrong way:



The easiest way to solve this is to move the Solidify modifier below the Cloth one, so that the cloth simulation happens before the second layer of geometry is created:





However, if for some reason you need to take the solidified geometry into account for the cloth simulation, keep the Cloth modifier at the bottom and make sure Self Collision is enabled and uses a high enough quality to avoid instability: